Points A, B, and C lie on a circle with center Q.
*The area of sector AQB is twice the area of sector BQC
*The length of arc AB is 28 centimeters
What is the length, in centimeters, of arc BC?
the area a is proportional to the arc length s.
s = rθ
a = 1/2 r^2 θ = 1/2 rs
so, if the area of BQC is 1/2 the area of AQB, BC is 1/2 of AB, or 14cm
Why did the circle go to the circus? Because it wanted to learn some new tricks! Now, let's get back to your question. We know that the area of sector AQB is twice the area of sector BQC. Since the area of a sector is proportional to the arc length, we can say that the length of arc AQB is twice the length of arc BQC.
If the length of arc AB is 28 centimeters, that means the length of arc AQB is also 28 centimeters. Since the length of arc AQB is twice the length of arc BQC, we can divide 28 by 2 to get the length of arc BQC.
So, the length of arc BC is 14 centimeters. Voila!
To solve this problem, let's denote the length of arc BC as x centimeters.
We know that the area of sector AQB is twice the area of sector BQC. The area of a sector is given by (θ/360) * π * r^2, where θ is the central angle in degrees and r is the radius of the circle.
Since the areas are proportional, the ratio of the central angles is the square root of the ratio of the areas:
(θ_AQB / 360) * π * r^2 = 2 * [(θ_BQC / 360) * π * r^2]
Simplifying this equation, we get:
(θ_AQB / θ_BQC) = 2
Now, let's work with the lengths of the arcs. The length of an arc in a circle is given by (θ/360) * 2 * π * r, where θ is the central angle in degrees and r is the radius of the circle.
Using this formula, we can write the following equation for the length of arc AB:
(θ_AQB / 360) * 2 * π * r = 28
Now, let's substitute the value of θ_AQB / θ_BQC from the first equation into the second equation:
[(2 * θ_BQC) / 360] * 2 * π * r = 28
Next, let's simplify the equation:
(4 * θ_BQC / 360) * π * r = 28
Let's cancel out the common terms:
(θ_BQC / 90) * π * r = 7
Finally, let's solve for the length of arc BC, denoted as θ_BQC / 360:
θ_BQC / 360 = (7 / (π * r)) * 90
θ_BQC / 360 = (630 / (π * r))
Now, we can substitute this value into the formula for the length of an arc in arc BC:
x = (θ_BQC / 360) * 2 * π * r
x = [(630 / (π * r)) / 360] * 2 * π * r
x = (630 * 2) / 360
x = 3.5
Therefore, the length of arc BC is 3.5 centimeters.
To find the length of arc BC, we first need to find the measure of the angle subtended by arc BC at the center Q. We can use the given information to solve this problem step by step.
Let's assume that the measure of angle AQB is x degrees. Since the area of sector AQB is twice the area of sector BQC, the measure of angle AQB is twice the measure of angle BQC.
So, the measure of angle AQB = 2 * measure of angle BQC
x = 2 * (360 - x)
x = 720 - 2x
3x = 720
x = 240 degrees
Since the length of the entire circumference of the circle is 360 degrees, we can set up a proportion to find the length of arc BC.
x degrees is to 28 centimeters as 360 degrees is to unknown length BC.
x/28 = 360/BC
Substituting the value of x we found earlier:
240/28 = 360/BC
Cross multiplying:
240 * BC = 28 * 360
Dividing both sides by 240 to solve for BC:
BC = (28 * 360)/240
BC = 42 centimeters
Therefore, the length of arc BC is 42 centimeters.